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A First Course in Probability

Sheldon M. Ross

$Math Studyset Vaia Explanations$ Math

9th Edition · 432 pages

ISBN: 9780321794772

Chapter 8: Q. 8.13 (page 393)

Show that if $E [X] < 0$ androle="math" localid="1649871241073" $θ \neq 0$ is such that $E [e^{θ X}] = 1$ , then $θ > 0$ .

Short Answer

Expert verified

Consider the function $f (y) = - \ln (y), y > 0$ . Since this function is convex, if we let $Y = e^{θ X}$ , using Jensen's inequality $E [f (Y)] \geq f (E [Y])$ ,

Step by step solution

01

Given Information.

$E [X] < 0$ and $θ \neq 0$ is such that $E [e^{θ X}] = 1$ , then $θ > 0$ .

02

Explanation.

Assume that $E [X] < 0$ and $θ \neq 0$ such that $E [e^{θ X}] = 1$ , whereby $θ$ is a scalar. We want to show that $θ > 0$ . To do this, we will use Jensen's inequality. It says that for a convex function $f$ and an arbitrary random variable $Y$ is

$E [f (Y)] \geq f (E [Y])$

Consider the function $f (y) = - \ln (y), y > 0$ . Since this function is convex, if we let $Y = e^{θ X}$ , using Jensen's inequality, we get:

$E [- \ln (e^{θ X})] \geq - \ln (E [e^{θ X}])$

Since $\ln (e^{u}) = u$ , for each $u$ , and using the information $E [e^{θ X}] = 1$ , we get:

$E [- θ X] \geq - \ln (1) = 0$

if properties of expectation

$- θ E [X] \geq 0$

$⇓$ it is given $E [X] < 0$

$- θ < 0$

$- θ < 0$

$θ > 0 .$

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Most popular questions from this chapter

It $X$ is a gamma random variable with parameters $(n, 1)$ , approximately how large must $n$ be so that $P \{|\frac{X}{n}| - 1 > . 01\} < . 01 ?$

Many people believe that the daily change in the price of a company’s stock on the stock market is a random variable with a mean of $0$ and a variance of $σ^{2}$ . That is if Yn represents the price of the stock on the $n$ th day, then $Y_{n} = Y_{n - 1} + X_{n}, n \geq 1$ where $X_{1}, X_{2}, . . .$ are independent and identically distributed random variables with mean $0$ and variance $σ_{2}$ . Suppose that the stock’s price today is $100$ . If $σ_{2} = 1$ , what can you say about the probability that the stock’s price will exceed $105$ after $10$ days?

A tobacco company claims that the amount of nicotine in one of its cigarettes is a random variable with a mean of $2.2$ mg and a standard deviation of $0.3$ mg. However, the average nicotine content of $100$ randomly chosen cigarettes was $3.1$ mg. Approximate the probability that the average would have been as high as or higher than $3.1$ if the company’s claims were true

Let $X$ be a binomial random variable with parameters $n$ and $p$ . Show that, for $i > n p$ ,

$(a)$ the minimum $e^{- t i} E [e^{t X}]$ occurs when $t$ is such that $e^{t} =$ $\frac{i q}{(n - i) p}$ where $q = 1 - p .$

$(b)$ $P {X \geq i} \leq \frac{n^{n}}{i^{2} (n - i)^{n - i}} p^{i} (1 - p)^{n - i}$

Suppose a fair coin is tossed $1000$ times. If the first $100$ tosses all result in heads, what proportion of heads would you expect on the final $900$ tosses? Comment on the statement “The strong law of large numbers swamps but does not compensate.”

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